Optimal. Leaf size=14 \[ x-\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {321, 212, 206, 203} \[ x-\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 212
Rule 321
Rubi steps
\begin {align*} \int \frac {x^4}{-1+x^4} \, dx &=x+\int \frac {1}{-1+x^4} \, dx\\ &=x-\frac {1}{2} \int \frac {1}{1-x^2} \, dx-\frac {1}{2} \int \frac {1}{1+x^2} \, dx\\ &=x-\frac {1}{2} \tan ^{-1}(x)-\frac {1}{2} \tanh ^{-1}(x)\\ \end {align*}
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Mathematica [A] time = 0.00, size = 26, normalized size = 1.86 \[ x+\frac {1}{4} \log (1-x)-\frac {1}{4} \log (x+1)-\frac {1}{2} \tan ^{-1}(x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 18, normalized size = 1.29 \[ x - \frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 20, normalized size = 1.43 \[ x - \frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} \, \log \left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 19, normalized size = 1.36 \[ x -\frac {\arctan \relax (x )}{2}+\frac {\ln \left (x -1\right )}{4}-\frac {\ln \left (x +1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 18, normalized size = 1.29 \[ x - \frac {1}{2} \, \arctan \relax (x) - \frac {1}{4} \, \log \left (x + 1\right ) + \frac {1}{4} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 10, normalized size = 0.71 \[ x-\frac {\mathrm {atan}\relax (x)}{2}-\frac {\mathrm {atanh}\relax (x)}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.13, size = 19, normalized size = 1.36 \[ x + \frac {\log {\left (x - 1 \right )}}{4} - \frac {\log {\left (x + 1 \right )}}{4} - \frac {\operatorname {atan}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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